Mutual Indwellinga. j . cotnoir

Perichoresis — sometimes translated ‘co-inherence’ or ‘mutual indwelling’ — is an important

concept in Christian theology, central to many historical and contemporary understandings of

the doctrine of the Trinity. The persons of the Trinity are said to indwell one another, and this

indwelling constitutes an intimate relationship between them. Perichoresis is used in different

theological frameworks to ground various other aspects of the doctrine of the Trinity, such as

the unity or oneness of the Trinity or the communication of attributes between the divine per-

sons. This notion of interpenetration or mutual indwelling can seem confusing or even perhaps

paradoxical, but I will suggest that a mereological understanding of this concept can serve to illu-

minate perichoresis. To be more precise: I argue that the relation of mutual parthood can provide

a coherent model of mutual indwelling of the persons on the Trinity.

The concept of mutual indwelling has its roots in the biblical texts, primarily in the Gospel of

John, where Jesus is quoted saying, “[…] you may know and understand that the Father is in me,

and I in the Father” (John :). Similarly, Jesus requests in John :-, “Believe me when I

say that I am in the Father and the Father is in me.”¹

Following others,² we may distinguish two versions of the concept: person-perichoresis and

nature-perichoresis. Person-perichoresis regards the perichoretic relation(s) that holds within the

Trinity between the three persons (Father, Son, and Spirit). Nature-perichoresis regards the peri-

choretic relation that holds within the hypostatic union between the two natures (human and

divine) of the incarnate Son.

As Crisp [, p. -] (following Prestige [] and others) notes, perichoresis was first

used by Gregory of Nazianzen in the th century A.D. followed by Maximus the Confessor. It

¹In fact, Jesus extends some form of mutual indwelling to his disciples when he promises the Spirit of truth who“will be in you” in John :-: “I am in the Father, and you are in me, and I am in you.” (Cf. John :-).

²Swinburne [, p. , n. ], Crisp [, p. ]

wasn’t until John of Damascus in the th century A.D. that the concept was applied to the Trinity

alongside the notion of ‘interpenetration’.³ By the time of the Council of Florence in A.D.

person-perichoresis had been fully incorporated into orthodoxy:

Because of this unity the Father is wholly in the Son and wholly in the Holy Ghost,

The Son is wholly in the Father and wholly in the Holy Ghost, the Holy Ghost is

wholly in the Father and wholly in the Son. ([, p. ])

But the concept of mutual indwelling is intuitively puzzling. According to all the usual ways

of conceiving of ‘indwelling’ or ‘being in’, it seems conceptually incoherent to think that two

persons could be in each other. If I am in my house, then my house is not in me. This just seems

to be a fact about the metaphysics of being in. The puzzling nature of mutual indwelling was

already made plain in the mid-th century A.D. by Hilary of Poitiers who writes:

[It] confuse[s] many minds, and not unnaturally, for the powers of human reason

cannot provide them with any intelligible meaning. It seems impossible that one

object should be both within and without another, or that […] these Beings can

reciprocally contain One Another, so that One should permanently envelope, and

also be permanently enveloped by, the Other, whom yet He envelopes. […] We

must think for ourselves, and come to know the meaning of the words, ‘I in the

Father, and the Father in Me’: but this will depend upon our success in grasping the

truth that reasoning based upon Divine verities can establish its conclusions, even

though they seem to contradict the laws of the universe. (De Trinitate, iii, , in [,

p. ])

The underlying conception does seem impossible, contradicting the metaphysical principles of the

universe. And yet despite its widespread use in theology (perhaps even its overuse⁴), the concept

³See Twombly [] for a full examination of John of Damascus on perichoresis.⁴cf. Kilby [], Otto []

of mutual indwelling has received comparatively little philosophical attention;⁵ and the puzzle

regarding its conceptual coherence is rarely examined.⁶ But without such analysis, there is the real

danger that an important theological concept be rejected as unintelligible, or perhaps dismissed

as ‘divine mystery’.⁷ But even the greatest and most difficult divine mysteries (e.g. the Trinity or

the hypostatic union) have been subject to unending philosophical analysis. We can and should

agree with Crisp [, p. ] that trying to better understand mutual indwelling is a worthwhile

enterprise, even if a complete understanding is “forever beyond us” (van Inwagen [, p. ]).

This article is an attempt to alleviate some of the puzzlement; I argue that a current formal

theory of an ordinary conception of ‘being in’ allows for mutual indwelling. In §, I examine

a range of possible proposals for modeling the perichoretic relationship, arguing that they will

not succeed. In §, I suggest a mereological way of understanding ‘being in’, by analogy with

independently motivated examples in contemporary metaphysics. Because I am understanding

perichoresis as a kind of parthood, it is crucial to show how such an understanding relates to the

doctrine of divine simplicity. In §, I argue that this proposal does not conflict with simplicity,

surprisingly, but instead affirms it. I conclude in §, by discussing how mutual indwelling relates

to the concepts of unity (inseparability) and identity (indiscernibility).

⁵The concept has seen use in philosophical theorizing on the Trinity, (e.g. Davis [, ], Hasker [, ch. ],Williams []), but almost nothing by the way of analysis.

⁶The situation leads Tuggy [] to complain:

But one suspects that all proponents mean by periochoresis in this context is ‘whatever it is which makesdivine persons combine to make a further person’. Even if it is true that the Father, Son, and Holy Spirit‘metaphysically interpenetrate’ (whatever this means) [...] such metaphors simply hide an unintelligibleclaim. ([, p. f ])

⁷Hilary of Poitiers seemed to think philosophical reflection would not be of any help:

This is a problem which the wit of man will never solve, nor will human research ever find an analogyfor this condition of Divine existence. But what man cannot understand, God can be. I do not mean tosay that the fact that this is an assertion made by God renders it at once intelligible to us. (De Trinitate,iii, , in [, p/])

Varieties of ‘Being In’

Several different philosophically robust concepts of ‘being in’ are available. I want to distinguish

two main groups: the predicational concepts and the containment concepts. Firstly, for the class

of predicational concepts, the relata of ‘being in’ are objects on the one hand and entities of a

higher ontological category on the other. A paradigm case is the instantiation relation holding

between an object and a property. This is a kind of ‘being in’, as when an attribute is ‘present

in’ the object it is true of. There is also the taxonomical relation of kind-membership, as when

something is ‘one of ’ or ‘among’ the Fs, for some suitable kind F. Similarly, there is the relational

‘in’, as when a two people are said to be ‘in love’ or ‘in a marriage’ by virtue of each of them

bearing the appropriate relation to one another.

Secondly, there is a class of containment concepts. Paradigm examples of a containment con-

cepts are set-theoretic such as ‘being a member of ’, or ‘being a proper subset of ’. Similarly, we

have plural locutions of ‘being one of ’ or ‘being a member of ’ a plurality of objects, as when I refer

to ‘one of my children’ or ‘a member of the Beatles’. Perhaps most centrally, we have locational

concepts of being in, involving an entity spatial location. Finally, there is also the mereological

conception of containment, namely parthood, as when an object is in another by being a part of

it.

One may attempt to understand perichoresis in terms of predicational conceptions of ‘being

in’. I won’t have much to say against such attempts as the main aim of this paper is to explore

to what extent we can make sense of mutual indwelling in terms of containment conceptions

of ‘being in’. But here is a reason to be dissatisfied with the predication conceptions. It is ini-

tially plausible that the person-perichoretic relation should be understood as holding between

the persons of the Trinity, who are themselves of the same ontological category. Predicational

conceptions of ‘being in’, however, are always trans-categorical — they hold between entities of

different ontological type.⁸ And this rules out a straightforward application of instantiation or

⁸This is clear enough when we are dealing with first-order entities like objects or persons and their properties whichare second-order; and this is all that really matters for the point being made here. Things are a little more contentious

kind-membership.⁹ That is, the persons are said to be in each other, they are not both (individ-

ually) ‘in F’ for some property or kind F. Similar considerations tell against the relational ‘in’:

it is by virtue of some third relation R between the persons (e.g. Heloise loving Abelard and

Abelard loving Heloise) that we say they (jointly) are ‘in R’ (e.g. ‘Heloise and Abelard are in love’

or ‘Heloise is in love with Abelard’ etc.).¹⁰ As a result, the relevant sort of ‘being in’ would not

a relation between the persons, but rather it would be relation between the persons and a some

relation in a higher ontological category.

None of this is dispositive against predicational concepts;¹¹ but I think it does provide a pro

tanto motivation for exploring containment conceptions thoroughly.

In the remainder of this section, I consider each of the containment conceptions of ‘being in’

that are best understood; indeed, each has a formally precise theoretical foundation. I begin by

arguing against set-theoretic and plural conceptions. I then consider the locational conception.

Finally, I then settle on the mereological conception as my key point of departure. I will argue

that there is a viable philosophical theory of parthood, complete with a coherent and consistent

formalization, that allows for mutual parts — distinct entities that are parts of each other. Such

a view can make sense of a perichoretic relation according to which it is absolutely coherent to

affirm that the three persons of the Trinity are ‘in’ the others.

. Containment

Set theory is the paradigm case of formally developed conception of ‘being in’. Sets are abstract

(mathematical) collections which have members or elements; elements of a set are in sets by virtue

when we are dealing with higher-level entities like properties and propositions, which presumably can themselves canhave properties. The standard view is that any properties of second-order properties must be third-order, and so of adifferent ontological kind. But of course the standard view might be rejected.

⁹But see §..¹⁰Linguistically speaking, the prepositional ‘in’ typically requires the nominalization of the given relation to serve

as a its complement. But this should not be taken to mean this sort of ‘being in’ is a relation between two first-orderobjects.

¹¹One might, for example, put forward a relational view according to which Jesus is ‘in union with’ the Father andthe Father is ‘in union with’ the Son. Here the relevant R is the perichoretic relation of unity itself, and the apparentsymmetry of ‘being in’ is explained away. I will not explore this proposal further; as my aim in this paper is to take thesymmetry of ‘being in’ seriously.

of the membership relation Pure sets have nothing but other sets among their members, or their

members’ members, etc. These are purely abstract. Impure sets can contain concrete objects

(called urelements) as members. Though they contain concrete urelements, impure sets are still

abstract.

This fact is a problem for modeling mutual indwelling: sets are abstract objects even when

its members are concrete. In the case of perichoresis, to model the claim that each person of the

Trinity is in the others, we would need to have it each person is an abstract object in order to be

capable of having members. But this would (should!) be objectionable to those who think God is

a concrete (even if non-material) entity.¹² Of course, it is not unusual for the purposes of model

theory to use set-theoretic entities and relations to represent concrete entities and relations; but

for our purposes (i.e. the philosophical explication of a puzzling concept) the explanatory value

of doing so seems dubious.¹³

The second main containment conception of ‘being in’ is found in plural logic. Pluralities

are somewhat like sets; they are governed by a membership relation ‘is one of ’: when x is one of

the Xs we can think of x as being contained in the Xs (e.g. Ringo is one of the Beatles). There

are a couple of main differences between sets and pluralities: firstly, pluralities are thought to be

concrete when their members are. So for our purposes, pluralities avoid the main problem with

sets. Secondly, pluralities are thought to be multiplicities whereas sets are thought to be singular

entities (e.g. my children are many; they are not some singular entity).

But this latter difference is also the main problem with plural understandings of mutual in-

¹²See Leftow [] for related discussion, albeit aimed at views in which essences are abstracta and God is identicalto God’s essence.

¹³In any case, there are other difficulties. The membership relation of Zermelo-Frankel set theory (ZF) is not a goodmodel for ‘being in’ because of the axiom of foundation which entails that there can be no cycles or loops in the setmembership relation. (See e.g. Theorem X in [, p. ].) Crucially it cannot be that X is in Y and Y is in X for anysets X and Y.

ZF and its descendants are not the only set theories, however. There are a number of set theories in which the axiomof foundation fails. Nonwellfounded set theories or hyperset theories replace the axiom of foundation with variations ofan Anti-Foundation Axiom which force there to be sets whose membership relation cycles back on itself (Barwise andMoss []). For example sets like A = {B} and B = {A} are perfectly consistent (Aczel []). Non-wellfounded settheories might provide a formal setting amenable to modeling a perichoretic relation of ‘being in’ — it may even showthat there is a coherent notion of ‘being in’ that allows for mutual indwelling. But given their exotic character and thelack of philosophical familiarity, doing so would serve little explanatory use.

dwelling. The claim that Son is ‘one of ’ the Father will only be coherent on the assumption that

the Father is a multiplicity — that he is a plurality. That’s intuitively and theologically problem-

atic, as the Father is an individual person.¹⁴ Moreover, the Father and the Son would both have to

be multiplicities (because the indwelling is mutual), and hence the ‘is one of ’ relationship would

not apply.¹⁵

The third main containment conception of ‘being in’ can be found in logics of location (Par-

sons [], Casati and Varzi []). Theories of location analyze the relation that objects have to

the spatio-temporal regions they occupy. Since regions may be contained in other regions, the

objects that have those locations may ‘inherit’ a kind of containment. So, one might define a

relation of ‘being in’ as follows: x is ‘in’ y if and only if the location of x is a subregion of the

location of y. This relation is perfectly compatible with mutual indwelling, since x is in y and y is

in x merely implies that x and y have identical locations, not that x and y themselves are identical.

So co-location could serve as a model of mutual indwelling.

This kind of containment solves the problems with previous notions; after all spatial contain-

ment relates one concrete object to another. However, if applied as a model of perichoresis, it

commits us to certain requirements on the locations of the persons of the Trinity that are contro-

versial. At the minimum, it requires that the persons of the Trinity have spatial locations, which

raises a host of controversial issues about God’s relation to spacetime. For example, Hasker [,

p. ] claims the divine persons cannot be co-located because God is non-spatial. By con-

trast, traditional considerations surrounding God’s omnipresence (Inman []) can yield further

¹⁴Perhaps, one could attempt to avoid this by arguing that each person of the Trinity is both one and many. In fact,some who endorse plural logic also argue additionally for many-one identities (see Cotnoir and Baxter []). Many-one identity might seem to be precisely the sort of view Trinitarians need (as argued by Bohn []), but it leads tounorthodox claims like ‘there is one God and many Gods’ (see Kleinschmidt [] for compelling arguments againstthis view). In any case, it seems as though we should be able to model perichoresis without being forced to the viewthat each person is a multiplicity.

¹⁵Of course we can generalize the one-many ‘is one of ’ relation to a many-many ‘are among’ relation defined asfollws: the Xs are among the Ys iff every member of the Xs is a member of the Ys (e.g. my children are among theschoolchildren). But this still won’t allow mutual amongness since the ‘among’ relation between pluralities is standardlythought to be antisymmetric: if the Xs are among the Ys and Ys and then Ys are among the Xs then Xs = Ys. That’sbecause, like sets, pluralities are individuated extensionally: the Xs and Ys are identical whenever they have exactly thesame members.

conflicts.

. Mereology

To sum up, we are after a kind of ‘being in’ relation such that the relata are neither abstracta

nor multiplicities. We want a containment relation which holds between a concrete object and

another unitary concrete object, and which doesn’t require spatial locations for these objects.

Fortunately, there is a formal theory of just such a relation — mereology.

Mereology is the formal theory of parts and wholes; it aims at a precise account of the parthood

relation. Here ‘parthood’ is to be understood as the general relation holding between entities and

their components, no matter what sort of things they are. For example, the keyboard is part of

my computer, my fingers are parts of my hands which are parts of me, the number ‘’ is part of

the function ‘x + ’. So, the relata of parthood can be either concrete or abstract.

There are a number of mereological theories in the literature, but according to so-called ‘clas-

sical mereology’, the parthood relation is what is known as a partial order. That is, the axioms of

this theory (typically) contain the following three principles governing parthood.¹⁶

Reflexivity For all x, x is part of x.

Antisymmetry For all x and y, if x is part of y and y is part of x, then x and y are identical.

Transitivity For all x, y, and z, if x is part of y and y part of z, then x is part of z.

Here I have followed standard procedure using ‘parthood’ as a relation that allows for identity to

be a limit case. The strict notion, incompatible with identity, is called ‘proper parthood’. Pretty

clearly, antisymmetry rules out mutual indwelling between distinct entities; it forces parthood to

be asymmetric in a way that is compatible with only the symmetries of identity.

Recently, however, the antisymmetry of parthood has come under fire. Philosophers have put

forward a number of purported counterexamples. Consider the Aleph, a strange object found in

¹⁶Of course there are other axioms too. I say ‘typically’ only because there are different axiomatizations of classicalmereology, some of which have these principles as theorems.

the basement of Beatriz Viterbo’s house in Borges’ story [, p. ]:

I saw the Aleph from all points. I saw the earth in the Aleph and in the earth the

Aleph once more and the earth in the Aleph ….

Sanford [] mentions this case to suggest that the naïve notion of parthood does not require

antisymmetry. As he notes, Borges uses the ‘in’ of containment and suggests that whatever the

Aleph contains is part of it. Consider a passage in the Upanishads that outlines the structural

relations between Brahman and persons:

In the center of the castle of Brahman, our own body, there is a small shrine in the

form of a lotus-flower, and within can be found a small space. […] This little space

within the heart is as great as this vast universe. The heavens and earth are there, and

the sun, and the moon, and the stars; fire and lightning and winds are there; and all

that now is and all that is not; for the whole universe is in Him and He dwells within

our heart. (Chandogya Upanishad, §., in [, p. ])

Other examples can be found in the from the Huayan school of Chinese Buddhism such as the

‘Net of Indra’ — a net of jewels that stretches infinitely in every direction, but in which each jewel

is contained in every other, symbolizing the interconnectedness of the universe.¹⁷

In the recent philosophical literature, a structurally similar example has been put forward by

Tillman and Fowler.

Suppose that the universe exists […] a thing such that absolutely everything is a

part of it. […] Assuming there is a unique such thing, let’s name it U. According

to a popular view of semantic content, ‘U exists’ semantically encodes a singular,

structured proposition that has U itself as a constituent as well as the property of

existing. By hypothesis, this proposition is a proper part of U. But U is in turn a

proper part of the relevant proposition. [, p. ]

¹⁷See Jones [] and Priest [, ch. ].

Tillman and Fowler take this as evidence that parthood must fail to be antisymmetric, a point

made also by Yablo [, p. ].¹⁸

A final example involves a time-travel case from Kleinschmidt [].¹⁹ Clifford is a dog-shaped

statue which was made partly of other statues, including Kibble—a small statue of a biscuit. But

Kibble, too, is made partly of other statues, including a time-traveling future version of Clifford

himself, suitably reduced in size. As a result, it would seem as though Kibble is a part of Clifford

and has Clifford as a part.

All of the above examples are somewhat exotic, requiring some dubitable metaphysical as-

sumptions (the possibility of time travel, the structured view of propositions with constituents

as parts). But it should be clear that some philosophers have rejected the ‘classical’ view that

parthood is a partial order. In the next section, I will claim that in fact there are many perfectly

ordinary cases of parthood that fail to validate antisymmetry. Indeed, there are viable mereological

theories that do not contain antisymmetry as an axiom or theorem.

An Analogy: the Statue and the Clay

To explain how a theory of parthood could allow for ordinary cases of mutual parts, I want to

draw on a standard example from contemporary metaphysics: the statue and the clay it is made

from. One major view among metaphysicians is that the statue and the clay are distinct entities

by virtue of their being discernible from one another. By ‘distinct’ here, I mean non-identical,

where identity is to be understood in the standard first-order sense. That is, identity is the unique

reflexive, transitive, and symmetric relation that satisfies the Indiscernibility of Identicals:

Indiscernibility of Identicals For all (qualitative) properties F, if x is identical to y, then Fx if

and only if Fy.

¹⁸Similar cases regarding parts of propositions are discussed in Cotnoir [], Gilmore [], and Merricks [, pp.–]

¹⁹Similar cases have been proposed and discussed in Daniels [], Eagle [], Effingham [], Effingham and Rob-son [], Gilmore [, ], and Smith [].

The clay and the statue are frequently thought to be discernible: for example, the clay but not

the statue can survive squashing; the statue might be Romanesque, whereas the clay is not; the

statue might be badly made where as the clay is not; the clay may be of poor quality but the statue

might be of very high quality, etc.²⁰ So we may conclude that they are non-identical.²¹

But though they are distinct, they still are completely spatially coincident and are made of

the same material parts. This conflicts with the controversial criterion for the identity of objects

called mereological extensionality according to which any two composite objects with all the same

proper parts are identical.

Classical mereology is extensional in precisely this way; it stipulates that objects are individu-

ated by their parts. If two objects are mereologically equivalent, then they are identical. As a result,

classical mereology forces on us the conclusion that objects which are mereologically indiscernible

are indiscernible tout court. That is a strong conclusion, indeed.

There are however formal mereologies in the recent literature which reject extensionality, and

they have recently been gaining traction.

. Two Kinds of Non-Extensional Mereology

Non-extensional mereologies fall into two kinds. The first are unsuppplemented mereologies,

which reject supplementation principles like the following:²²

Weak Supplementation If x is a proper part of y, there is some part of y that is disjoint from x.

(Here, by ‘disjoint’ I mean that they do not overlap, i.e. the two objects have no parts in common.)

This principle gives voice to the intuitive thought that if an object has a proper part it should also

have some other part, particularly one which does share any parts with the other. Supplementation

principles are extremely plausible, and play a number of important roles in formal mereology.

²⁰See e.g. Johnston [] and Fine []. For an overview of such arguments, and relevant references, see [],Magidor [], Paul [], and Blatti [].

²¹Whether such properties that discern the statue and clay are themselves qualitative is controversial.²²See e.g. Gilmore [].

The second kind of non-extensional mereology are mutual parts mereologies; they reject the

antisymmetry principle for parthood. On such a view, the statue and the clay, though distinct,

are part of each other. This view was first defended by Thomson [, ] and more recently de-

fended by Cotnoir []. The statue and the clay, and any other cases of mereologically coincident

objects, are perfectly ordinary, non-exotic cases of mutual parthood.

Given the antecedent plausibily of non-extensionalism, it is worth taking the time to make

the case that mutual parts mereologies have advantages over unsupplemented mereologies for

this purpose. Sider [, p. ] argues that if the statue and its matter share the same parts then

they must be mutual parts. His argument relies on another mereological principle, called Strong

Supplementation.

Strong Supplementation If x is not part of y, there is some part of x that is disjoint from y.

Since the statue and its clay share all their parts in common, there cannot be a part of one that

does not overlap the other, and hence by (the contrapositive of ) the above principle, both are

parts of one another.

Recently, other arguments have been put forward in favor of mutual parthood anti-extensional

mereology. Cotnoir [] puts forward a number of arguments against unsupplemented mereolo-

gies by outlining the key roles supplementation principles play in formal theories of mereology,

in addition to arguing that a mutual parts mereology is the best anti-extensional mereology com-

patible with universalism.

There is a sense in which antisymmetry just is an extensionality principle for parthood. Com-

pare three main forms of extensionality:

Extensionality of Proper Parthood If x and y are composite objects with all and only the same

proper parts, then x = y.

Extensionality of Overlap If x and y overlap all and only the same things, then x = y.

Extensionality of Parthood If x and y have all and only the same parts, then x = y.

All such principles assert an identity between objects that share certain mereological properties.

Antisymmetry has the same form, asserting an identity between mutual parts. In fact, if a is part

of b and b is part of a, by transitivity we know that every part of a is part of b and vice versa. Hence,

in the presence of transitivity, antisymmetry is equivalent to the extensionality of parthood.

In short, anyone who is sympathetic to non-extensionalism for material objects ought to take

mutual parthood mereology as a serious contender. And since parthood is one sort of containment

relation between objects (including concrete objects), it is a good candidate for modeling mutual

indwelling.

. A Formal Theory of Mutual Indwelling

Here then is an axiomatization of a mereology that allows for mutual parthood.

Reflexivity For all x, x is part of x.

Transitivity For all x, y, and z, if x is part of y and y is part of z, then x is part of z.

Strong Supplementation If x is not part of y, then there is some z that is part of x that is disjoint

from y.

Composition For any Xs if φ holds of the Xs then there is some z such that z is a fusion of the

Xs.

The final axiom of Composition is included for completeness since mereological theories typical

include axioms that characterize under what conditions composition occurs. Here, I have simply

used a schema, where φ here can be any condition one likes: e.g. if φ is just the condition that

there is at least one of the Xs then the result will be mereological universalism, or φ might be the

more restrictive condition that the Xs are structured in a certain way, etc. Also, ‘z is a fusion of

the Xs’ means ‘for all y, y overlaps z if and only if y overlaps one of the Xs’.

This theory has many models that allow for mutual parthood. We can draw these models

using directed graphs, with letters at each node representing an object and arrows representing

the parthood relation. (Chaining together arrows also counts as an instance of parthood.) A

model of the statue and the clay, for example, might look like this:

a ** bjj

c

OO 88rrrrrrrrrrrrr d

OOffLLLLLLLLLLLLL

Here a and b are the statue and clay respectively, and c and d parts of both.²³

Here is another model, this time of three objects, all of which are mutual parts of one another:

a

����b **

AA

cjj

SS

This last model is particularly salient for our purposes, since it purports to have the structure

of the perichoretic relations between the persons of the Trinity. In each case, every person is part

of another person who is in turn part of the first. We can see that every person is a mutual part

of every other person of the Trinity.

These models show that the formal system is logically consistent; there is nothing contradic-

tory about a relation that satisfies these axioms. And insofar as we are willing to accept that this

axiomatic system represents a kind of containment conception of ‘being in’, we have shown that

mutual ‘being in’ is formally coherent.

Divine Simplicity

In modeling perichoresis as a kind of parthood relation, the proposal would seem to run the risk

of conflicting with the long-standing albeit controversial doctrine of divine simplicity. In this

section I will outline six different versions of the doctrine of simplicity, and address each of them

²³For convenience we are leaving aside the obvious possibility that c and d have other parts.

in turn. I’ll argue that on each construal, God can be considered to be simple even if we accept

the mutual parthood model of perichoresis.

The doctrine of divine simplicity has too long of a history to outline here. Its foremost recent

critic is Plantinga [] (see also Mullins [], Schmitt []). But the doctrine also has many

recent proponents (Brower [], Davies [], Dolezal [], Leftow []). There are a number of

different things one might mean by this doctrine, however. Consider:

. God is simple by virtue of being immaterial; he has no material parts.

. God is simple by virtue of transcending spacetime; he has no spatial or temporal parts.

. God is simple by virtue of being identical to his attributes; he has no formal or qualitative

parts.

. God is simple by virtue of being undivided; his divinity is not split between the divine

persons.

. God is simple by virtue of being independent; he is not composed of anything more fun-

damental than himself.

. God is simple by virtue of having no (proper) parts of any sort.

Each of these doctrines are distinct, with () being the most general formulation. I want to argue

that all of them may be affirmed by those who accept our model.

Starting with (), note that in modeling perichoresis as a parthood relation I have not made

any assumption that God is made of matter nor that the parthood relation is restricted only to

material objects. Regarding (), given that I have not analyzed parthood in terms of spatial or

temporal relations, there is no assumption here that God has spatial or temporal parts. Of course,

spatial intuitions often drive judgements about ‘containment’ or ‘being in’, but it needn’t be so

(compare the examples in §.).Nothing about this analysis entails that all parts must be located

in spacetime, and so there is no reason to think the view violates simplicity of type ().

As for (), we come to the most controversial doctrine of simplicity: God’s identity with his

attributes. This version of simplicity is based on a background constituent ontology according to

which attributes are themselves parts of the concrete particulars that instantiate them. Contrast

relational ontology according to which attributes are not parts of the concrete things that instan-

tiate them.²⁴ If a constituent ontologist were to accept that God’s attributes are distinct from

himself (or from one another) then God would have distinct parts and hence would be metaphys-

ically complex, violating simplicity. Relational ontologists can quite happily accept that God has

no formal or qualitative parts, as they already accept that nothing concrete has such parts.²⁵ As

far as I can see, nothing about modeling perichoresis as a kind of parthood conflicts with (), and

so I will leave this thorny issue aside.

Now we turn to ()–(), the versions of divine simplicity that most directly threaten the

current proposal. I will address each question in turn.

. Distinction without Division

The question raised by () is whether God is rightly thought to be divided or partitioned into

the divine persons; the answer within classical theology has nearly always been in the negative.

Consider the High Medieval theologian Peter Lombard:

Nor is any of the three persons a part of God or of the divine essence, because each of

them is truly and fully God and is the whole and full divine essence and so none of

these persons is a part of the Trinity. (Sententiarum, I, xix, ; in [, vol. , p. ])

The idea here is that each person of the Trinity should not be thought of as a part of God, since

in the ordinary setting that would imply that each person of the Trinity is not wholly or fully

God. This is intuitively correct, as it appeals to the thought that where there are proper parts, there

²⁴The connection between constituent and relational ontologies and the doctrine of simplicity was made first byWolterstorff []. For more details on the metaphysics, see van Inwagen [].

²⁵This is not to say that relational ontologists don’t have any problems in giving an account of God’s attributes, butwhatever those problems are, they are not mereological.

is a remainder. (Recall this is the same natural thought that lies behind various supplementation

principles.) God is not divided in this sense.

In modeling perichoresis as a kind of parthood, one might wonder whether I have run afoul

of this idea. Although the model only claims that the three persons of the Trinity are parts of

each other, we also want to affirm that each person is God. Does this mean that God is divided

amongst the persons? No. After all, each person is itself ‘included in’ every other person. When

it is claimed that the Son is God, it is simply false that there are other parts of God not included

within the Son. And this is precisely what our model of mutual indwelling secures. The Son is

fully God and the fullness of God is ‘in’ him; that is, the entirety of the Trinity dwells in him.

Here is another way of illustrating the point. Consider again the relevant model of perichore-

sis:

a

����b **

AA

cjj

SS

Which of the elements of this model is composed of the others? The answer is complicated by the

fact that the composition relation here is not unique — that is, there are Xs such that more than

one thing that counts as being a fusion of the Xs. In fact, since everything overlaps everything

else in this model, every person counts as a ‘fusion’ of every collection of persons. The Father

is composed of the Father, Son, and Spirit; the Son is composed of the Father, Son, and Spirit;

and the Spirit is composed of the Father, Son, and Spirit as well. Because they all mutually

interpenetrate one another, they are each ‘composed’ of all the others.

This situation should be sharply contrasted with another mereological model with three basic

elements:

d

a

GG�������������b

OO

c

WW/////////////

This model is strikingly different in that each of the three basic elements are composed into a

fourth thing. That fusion is unique, and has each of its basic elements as proper parts. In this

case, some fourth thing d is divided among three others a, b, c; in this case, because d has a as

a proper part, there is some remainder of d that isn’t included in a. This is not the model of

the persons of the Trinity I am proposing; let me emphasize — I am not proposing that God is

composed of the three persons of the Trinity, each of which is a proper part of God. To mistake

this model for the one above would be to misunderstand the main contention of this paper:

the mereological relation of mutual parthood can provide a coherent model of perichoresis; the

mereological relation of composition, however, does not account for the unity or oneness of God.

. Dependence and Mutual Parthood

Now to the question raised in (): is God composed of anything more fundamental than himself?

Consider Anselm:

For, everything which is composite requires for its subsistence the things of which it

is compounded, and, indeed, owes to them the fact of its existence, because, whatever

it is, it is through these things. (Anselm, Monologion XVII, in [])

And Aquinas:²⁶

On the contrary, every composite is posterior to its components: since the simpler

exists in itself before anything is added to it for the composition of a third. But

²⁶Compare Summa Theologica .. []: “Every composite thing is posterior to its components and dependent onthem. But, as was shown above, God is the first being.”

nothing is prior to the first. Therefore since God is the first principle, He is not

composite. (Aquinas, Scriptum super libros sententiarum I... in [, p. ])

The reasoning here regards God’s ontological priority over all other things. One of the main

motivations for adhering to divine simplicity is aseity, the thought that God is not dependent on

anything for his existence. It is frequently thought that composite objects are dependent on their

parts for their existence. So God must be simple, lest he be dependent on his parts.²⁷

This argument for simplicity depends, of course, on the independent premise that wholes

depend or are grounded in their parts — a premise which has recently become extremely con-

troversial. In recent metaphysics it has come under heavy attack on a number of independent

grounds. Schaffer [] has put forward a number of arguments for priority monism — the view

that the whole universe is ontologically prior to its many parts — on the basis of physical²⁸ and

metaphysical²⁹ considerations. Priority monism is a radical inversion of the usual part-whole de-

pendence order. But recently it has been put in service of the doctrine of the Trinity to preserve

aseity without simplicity (Fowler []).

Rather than purely inverting the order of dependence, others have simply denied that depen-

dence and parthood correlate at all. Some argue that wholes depend on their parts whilst other

wholes are prior to their parts;³⁰ others suggest that the dependence can run either up or down

the mereological hierarchy depending on the kind of part it is.³¹ Johnston [] is explicit about

the implications for divine simplicity:

In the tradition of Classical Theism, a paradoxical consequence has been drawn from

this, namely that God is utterly simple, so that his essential attributes must all be

identical, and be identical with him. […] But these paradoxical conclusions only

²⁷See Brower [] and Morris [] for more on the relation between the doctrines of divine simplicity and aseity.²⁸See also Calosi [].²⁹See also Cameron [], Esfeld [], Morganti [], and Trogdon [].³⁰See e.g. Inman [] and McDaniel [, p. f ].³¹See McDaniel’s [] discussion of Armstrong, and especially the neo-Aristotelians e.g. Fine [, §] and John-

ston [, §] who think that wholes depend on their ‘constituent’ or ‘structural’ parts (like the mechanical parts of amotorcycle), but whose ‘decompositional’ parts (like the Northern or Southern hemisphere) depend on the whole. Seealso Koons [] and Marmodoro [].

follow given Thomas’s false assumption that every composite thing is posterior to its

components, and dependent on them. ([, p. ])

So there is generally independent reason to doubt the crucial premise in the main argument for

divine simplicity. A mereologically complex God need not be dependent on anything.

Such a defense is available to the view of mutual indwelling developed here. There are really

two main options for those who accept a mutual parts mereology. The first is simply to deny that

ontological dependence tracks mereological parthood. In fact, insofar ontological dependence

is a kind of metaphysical explanatory relation, we may have even more reason to be suspicious.

Mutual parts would entail a cycle of metaphysical explanation — a whole grounded in a part that

is grounded in that whole — which may well be regarded as viciously circular (Lowe [, ]).

A second option would be simply to accept the correlation between parthood and ontological

dependence, and simply accept that ontological dependence is not itself asymmetric (a principle

which has itself recently become controversial viz. Barnes [], Rodriguez-Pereyra []). If cases

of mutual parthood are cases of symmetric dependence, one might well agree with Bliss []

and Thompson [] that symmetric dependence is not viciously circular. In the case of the

Trinity, then, God would be a paradigm case of an ontologically interdependent entity without

this violating God’s aseity. Everything that is not God is dependent on God, whilst God is not

dependent on anything that is not God, even if there can be mutual interdependence within the

Trinity.³²

Evaluating the metaphysical and theological merits of each response is for another time (though

I prefer the first on metaphysical grounds). If either is successful, though, there is no deep conflict

between my model of mutual indwelling and the the aseity of God.

³²Some language used in describing the relations between the persons of the Trinity like ‘begets’ and ‘proceeds from’suggests dependence relations in the Trinity. I do not think of the divine processions in this way, but how such languagerelates to aseity and ontological dependence is beyond the scope of this paper.

. Two Definitions of Proper Parthood

Let us turn now to the more general doctrine of simplicity raised in () — the claim that God has

no proper parts of any sort. (Since God is self-identical he is an improper part of himself; so ()

must be phrased using the notion of proper parthood.) But how exactly should we understand

proper parthood?

In mereology, there are two candidate definitions for proper parthood.

Proper Parthood (PP) x is a proper part of y iff x is part of y and x is not identical to y

Proper Parthood (PP) x is a proper part of y iff x is part of y and y is not part of x

While the former definition PP is presumably more common, many authors have relied on the

identity-free version PP, following Goodman [].³³ In classical mereology, the two definitions

are equivalent: PP logically entails PP, but the converse entailment requires antisymmetry. So

in the context of a mutual parts mereology the equivalence breaks down, and there are compelling

reasons to prefer the second definition.

The first reason in favor of PP as the correct definition for proper parthood concerns the

transitivity of proper parthood. Proper parthood should be transitive if parthood is, but this

occurs only on PP. PP allows violations of transitivity, as in the following countermodel.

a ** bjj

If a is a proper part of b and b is a proper part of a, it will not follow that a is a proper part of a

as that would contradict PP.

A second reason to favor PP as the correct definition for proper parthood is that it, but not

PP, is compatible with Supplementation. Recall (from §.) that any object that has a proper

part must have another part disjoint from the first. The principle is extremely intuitive; so natural

³³Examples include Casati and Varzi [, p. ], Eberle [], Niebergall [, p. ], [, p. ], and Simons [,p. ].

in fact that many authors have gone so far as to call it analytically true.³⁴ But it would be false

on the PP definition: the same countermodel as above can be used. If a and b are mutual parts,

then a is a proper part of b on the PP definition. But by transitivity every part of b is a part of a

and hence there can be no proper parts of b that do not overlap a. This sort of situation does not

arise with PP, however, since no mutual parts are ever proper parts on the PP definition.

This last fact is important, since according to the PP definition an entity that has nothing

but mutual parts will have no proper parts. To be a simple is to have no proper parts, and hence

entities like a and b in the above model count as simples. The same holds true, of course, in the

three-element model of perichoresis. Each person of the Trinity has no proper parts (given the

only plausible definition), and hence each is mereologically simple.³⁵

One might object: doesn’t this accommodation of divine simplicity undermine our original

claims that the Father is in the Son and the Son is in the Father? Can’t we make a similar distinction

between ‘being in’ and ‘being properly in’ on analogy with the PP definition? And if we do,

wouldn’t it follow that the Father is not properly in the Son, and the Son is not properly in the

Father?

In short, yes, these definitions are available and they do have these consequences. I do not agree

that this undermines our aims, however. Claims of ‘being properly in’ were not the perichoretic

claims to be modeled. Moreover, we should not affirm that the Father is properly contained in

the Son, nor vice versa, precisely because we do not have the requisite supplementation intuition

regarding the Trinity — there is no ‘remainder’ of the Father without the Son, nor is there any

³⁴Varzi [, p. ] claims ‘This principle expresses a minimal requirement which any relation must satisfy […] ifit is to qualify as parthood at all.’ Simons [, p. ] claims that ‘[Supplementation] is indeed analytic—constitutiveof the meaning of proper part’. Similar claims to analyticity are endorsed by Bohn [, p. footnote ], Koslicki [,p. –], and McDaniel [, p. ].

³⁵This raises the question of whether any person of the Trinity has any non-mutual parts; for example, an anonymousreferee wondered whether the Son has his human nature as a proper part. As I think of natures as properties, and becauseI reject constituent ontology, the human nature of Jesus is not one of his parts. What about other material parts, e.g.the hands and feet of Jesus? These bodily parts should count as proper parts in the ordinary sense, but it’s not clear ingeneral how material parthood relations within the body of Jesus as incarnate Son will integrate with claims about theTrinity. For example, we wouldn’t want to conclude that God has material parts from the claims that Jesus is God andJesus had material parts. Nor will we want to conclude that God has proper parts mutatis mutandis. In any case, theseare interesting issues I hope to write about in future work.

‘remainder’ of the Son without the Spirit, etc.³⁶ Mutual indwelling, on the mereological model,

allows for parthood among distinct individuals without division or remainder.

A Disanalogy: Distinctness and Separability

As a model, I think the mereological conception of perichoresis can do some important philo-

sophical and theological work. However, as with any model, there will be disanalogies. In this

section I discuss one particular disanalogy between the motivating examples (e.g. the statue and

the clay) and mutual indwelling of the divine persons related to the unity of the Trinity. I con-

clude by considering some ways of accounting for this unity, including modal inseparability and

constitution.

The key disanalogy concerns the whether the persons of the Trinity are separable, i.e. whether

any of the persons can exist without any of the others. Augustine thought this inseparability was a

straightforward consequence of simplicity: “It is for this reason then that the nature of the Trinity

is called simple, because it has not anything which it can lose […]” (Augustine, City of God, Bk.

, in [, p. ]). Even in the earliest uses of person-perichoresis for the doctrine of the Trinity,

the persons were thought to be inseparable.

The abiding and resting of the Persons in one another is not in such a manner that

they coalesce of become confused, but, rather, so that they adhere to one another,

for they are without interval between them and inseparable […] ” (John Damascene,

The Orthodox Faith, I..-, in [, p. ])

Now, there are a number of different kinds of inseparability: it may be humanly impossible to

separate two magnets of a certain strength, or it may be physically impossible to separate time and

space, or it might be metaphysically impossible to separate the triangular from the trilateral, or

³⁶If one really wanted to insist that proper indwelling is the correct perichoretic notion, there are at least two avenuesto capture that thought. First, one could simply opt for the PP notion of proper parthood, and would then have toreject transitivity and supplementation (which shouldn’t hold in the case of the persons of the Trinity anyway). Second,one could follow Cotnoir and Bacon [] and axiomatize proper parthood directly as merely a transitive relation. Onthis route, only Strong Supplementation can be maintained, and entities can be proper parts of themselves.

logically impossible to separate the truth of a proposition from the truth of its double-negation.

The key question for separability is this: to what extent is it possible (on some understanding of

the relevant modality) for one to exist without the other?

The persons of the Trinity, as co-eternal, are temporally inseparable; similarly, as metaphys-

ically necessary they are metaphysically inseparable. By contrast, the statue and clay are clearly

separable: one can exist without the other. The clay is typically made before a statue. There might

be worlds like ours in which the very same clay exists but no statue was made, or worlds very

similar to ours in which that particular portion of clay didn’t exist and so the same statue was

made from a different portion. As a result, three persons of the Trinity are more strongly united

than the statue and the clay, and hence mutual parthood is not sufficient to account for this unity.

This is something of a drawback for the model, as perichoresis is typically intended to function

as a means of uniting the persons of the Trinity. Of course, one can accept the metaphysical

impossibility of their separability, by dint of their necessary existence. But this inseparability is

not due to perichoresis; their unity is not explained by their mutual indwelling.

There are some kinds of unity that can be accounted for by parthood, as parthood is a signif-

icantly intimate relation. Sider [, p. ] outlines two key ways in which that intimacy shows

up.³⁷

Inheritance of Location If x is part of y, then y is located wherever x is located.

Inheritance of Intrinsicality If property P is intrinsic, then the property having a part that has

P is also intrinsic.

The Inheritance of Location entails mutual parts will have to be co-located; and this corresponds to

a robust notion of spatial inseparability. Of course this only holds for parts that have locations, and

I’ve already questioned whether this is appropriate for the divine persons in §.. The Inheritance

of Intrinsicality entails every intrinsic property of a part (‘P’) will correspond to an intrinsic

property of the whole (‘having a part that has P’), and for mutual parts vice versa. The import of

³⁷See also Gilmore [] for discussion.

this inheritance principle for the divine persons is that the intrinsic properties of each person of

the Trinity (e.g. ‘being begotten’) correspond to some other property (e.g. ‘being such that the

begotten dwells within me’) intrinsic to every other person of the Trinity.

While these notions of intimacy and unity may well be philosophically and theologically

important; it should be clear enough that they cannot carry the whole weight of the unity of the

Trinity.

. Constitution and Separability

There is at least one other sense in which mutual parts are united that is worth highlighting: in

the usual material examples, mutual parts are united by being made of the same matter.

One main rival to the mutual parthood view makes heavy use of this shared matter: it’s the

view that the statue is constituted by the clay. Many, following Aristotle, hold to a hylomorphic

view of material objects such that the matter (e.g. the clay) is part of the whole (e.g. the statue),

though not vice versa.³⁸ Such a view accepts the numerical distinctness of the statue and the

clay, but they do not typically accept anti-extensionalism.³⁹ That is because hylomorphic views

generally accept that there are parts (or perhaps ‘constituents’) of the statue, namely its form, that

distinguish it from the clay.

To be clear, this is not the view I am defending here. I do not accept that there are formal

parts (or constituents) that serve to mereologically distinguish the statue and the clay; they are

distinguished by their properties and properties are not parts. Nor do I here undertake a com-

mitment to a basic metaphysical relation of constitution that holds asymmetrically between the

statue and the clay.

The constitution view of material objects has been put forth (by Brower and Rea []) as a

model of the metaphysics of the Trinity.⁴⁰ Imagine a clay statue is used as a pillar in a cathedral.

³⁸See Fine [], Haslanger [], Koslicki [], and Rea [, ].³⁹Some theorists accept a constitution relation without understanding constituting matter as a part. See e.g. Baker [,

].⁴⁰See also Hasker [, ch. ] for a somewhat different constitution model.

Just as the statue and pillar are numerically distinct but made of the same material stuff (the

clay), so too can the Father, the Son, and the Spirit be numerically distinct but made up of the

same immaterial stuff (the divine essence). But just as the statue and the pillar are (in some sense

or other) ‘the same material object’, so too are the Father, Son, and Spirit ‘the same immaterial

object’. Instructive as it may be, the constitution view of the Trinity has faced a number of

criticisms.⁴¹

From the perspective of the aims of this paper, it is not immediately obvious how the con-

stitution model can make sense of mutual indwelling. Though the persons are the ‘same’ in an

important sense, there is no clear sense in which they are ‘in’ one another. One potential avenue

is as follows: one could make use of a predicational concept of ‘being in’, in particular the relation

of instantiation. Each person of the Trinity instantiates the divine essence, and hence this essence

is ‘present in’ each person. By itself, this doesn’t quite deliver mutual indwelling between the

persons, but only a shared essence. To supplement, one might suggest that a kind of systematic

ambiguity thesis: so when Jesus says “I am in the Father and my Father is in me” what is meant

is that “I [my essence] is in the Father [the person] and the Father [his essence] is in me [the

person].” This delivers an interpretation of the target perichoretic claims, but seems to me to be

something of a stretch, however. Alternatively, one could accept (perhaps by divine simplicity)

that each person is identical to the divine essence. If essences are still said to be ‘present in’ the

persons they are identical to, this would deliver the truth of the perichoretic claims. In this case,

though one wonders how material constitution sheds any light on the underlying metaphysics.

While I think Brower and Rea are right to look for some kind of sameness without identity, I

do not think constitution is the right sort of notion. Sameness of matter (or ‘immaterial stuff’)

is not sufficient for two things to count as the same material object, because it is not sufficient

for inseparability. The statue and the pillar are clearly separable: one can exist without the other

temporally and modally, even though they are constituted by the same matter. The statue might

be removed from the plinth in the cathedral and placed in a museum, thus ceasing to be a pillar

⁴¹See Craig [], Hasker [], and Tuggy [].

whilst remaining a statue. Or, the sculptor might be unhappy with the shape of statue, and so

take it down and reform the very same clay into a pure cylindrical pillar, whereby the statue

ceases to exist. It seems natural to think that the degree to which one can exist without the other

informs our judgements about whether one thing is the same object as another. So in what sense

do they count as the same material object?⁴² In sum, neither constitution nor mutual parthood

is sufficient to account for this deep metaphysical unity.

It is worth contrasting the aims of this paper with the aims of Brower and Rea []. They set

out to provide a metaphysically perspicuous model of ‘sameness without identity’ to help explain

how the three persons of the Trinity can be one God. My aims were much more modest; I didn’t

set out to solve all philosophical difficulties with the doctrine of the the Trinity. I merely wished

to solve the puzzle as to how any reasonable conception of being in could allow us to make sense of

mutual indwelling. In my view, Brower and Rea are right that the puzzles of material constitution

can illuminate the Trinity, but not in the way that they suppose.

. Unity and Oneness

I still have not delivered a fully adequate model of divine unity; and as a result, we have not given

a full account of the doctrine of the Trinity. We should not rest content with a mere ‘social’ model

of three persons that mutually indwell one another; critics of social trinitarianism may well be

right that perichoresis alone cannot account for the unity of the three persons. The doctrine of

the Trinity requires us to make sense of a deep metaphysical oneness of God. After all, almost

immediately after Jesus expresses his mutual indwelling with the Father, he claims “I and the

Father are one” (John :). In the words of Augustine:

There are the Father, the Son, and the Holy Spirit, and each is God and at the same

time all are one God; and each of them is a full substance, and at the same time all

⁴²Of course this sort of coming-into and going-out-of existence is impossible with the Father, Son, and Spirit as theyare (arguably) necessary existents. But that can’t be the whole story of their unity; after all, we do not think of, say, thenumber and the Father as united simply because they are both necessary existents.

are one substance. (Augustine, On Christian Doctrine I.., in [, p. ])

I have been assuming throughout that the persons of the Trinity are distinct, i.e. non-identical.

This is in part because trinitarian monotheism requires one to affirm that there are three divine

persons. But it also follows from the Indiscernibility of Identicals: there are truths about each

person that are not true of the others. So we cannot, I think, ensure the oneness of the persons by

identity.

What is required, then, is a metaphysics of unity that is not tied to qualitative indiscernibil-

ity.Of course, this is nothing more than a pointer in a certain direction. A fully worked out model

of the doctrine of the Trinity is a long way off. But providing a conceptually coherent model of

mutual indwelling is a good start.⁴³

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